✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 40² = ?
- √2025 = ?
- Kaunsa bada: 15² ya 230?
Answers dekho
- 1600
- 45
- 15² = 225, to 230 bada
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 4 — Square Root Kya Hai? aur Post 2 — Square Numbers Ki Properties ka aakhri digit table dekh lijiye.
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🎬 Kahani: Aribam Aur Bijou Ka Root Game
Aribam aur Bijou ek game khel rahe the: ek number bolta, doosra uska square root. “25!” “5!” “81!” “9!” Phir Aribam bola: “250!” 😈
Bijou atak gaya: 250 perfect square nahi! Aribam: “Theek hai, kam se kam paas ka number to batao.” Aur doosri taraf Akhil ke paas 125 sq cm ka square kapda hai: kya 15 cm ka rumaal katega? Dono ko andaaza chahiye.
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📖 Concept: Do Squares Ke Beech Ghero, Phir Pass Karo
💡 Is post ki ek badi baat: Root ka andaaza = number ko do nazdeeki perfect squares ke beech gherna: a² < y < b² to a < √y < b. Phir aakhri digit se options kam karo aur beech ka square (jaise 45²) nikaal ke interval aadha karo.
Step 1: Pehla ghera: das ke squares
√1936? 40² = 1600 < 1936 < 2500 = 50² → 40 < √1936 < 50.
WHY? Bada number = bada root. Square badhta to root bhi badhta, isliye ghera sahi.
Step 2: Aakhri digit se do options
1936 ka aakhri digit 6 → root 4 ya 6 pe khatam (4² = 16, 6² = 36). Options: 44 ya 46.
Table: square 1 → root 1/9; 4 → 2/8; 9 → 3/7; 6 → 4/6; 5 → 5; 0 → 0.
Step 3: Beech ka square: interval aadha
45² = (40 + 5)² = 1600 + 400 + 25 = 2025 > 1936 → root 40 aur 45 ke beech → 44.
Check: 44² = 1936 ✅ (Shortcut: 5 pe khatam ka square a(a + 1)|25, Post 2.)
Step 4: Jo perfect square nahi: √250
10² = 100, 20² = 400 → 10 < √250 < 20. Paas aao: 15² = 225, 16² = 256 → 15 < √250 < 16.
250 se 256 sirf 6 dur, 225 se 25 dur → √250 ≈ 16 (thoda kam). Asli ≈ 15.8.
Step 5: Rumaal ka sawaal (integer side)
Kapda 125 sq cm. 15 cm ka rumaal = 225 sq cm > 125 → nahi katega.
Sabse bada integer side: 11² = 121 ≤ 125 < 144 = 12² → 11 cm. Yahan “paas wala” nahi, neeche wala chahiye, kyunki kapda kam nahi pad sakta.
🪜 Atak Gaye? Aise Socho
- Pehle 10, 20, 30… ke squares se ghero.
- Phir ek-ek se paas: 15², 16²…
- Perfect square hai? Aakhri digit se 2 options.
- Beech ka 5 wala square nikaal ke ek option hatao.
- Sawaal “sabse bada fit” poochhe to neeche wala integer lo.
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🔐 Deep Concept: √2 Ka Andaaza Aur Decimal Tak Pass Jaana
1² = 1 < 2 < 4 = 2² → 1 < √2 < 2. Ab decimal: 1.4² = 1.96, 1.5² = 2.25 → 1.4 < √2 < 1.5. Phir 1.41² = 1.9881, 1.42² = 2.0164 → √2 ≈ 1.41.
Gehri baat: har kadam pe ghera 10 guna chhota. Yeh “ghero aur pass karo” soch computers bhi use karte hain. √2 kabhi khatam nahi hota (na integer, na fraction), par andaaza jitna chahe utna pass.
Kab kaunsa andaaza? “Lagbhag kitna” poochha → nazdeeki integer. “Kitna fit hoga” → neeche wala. “Kam se kam kitna chahiye” → upar wala.
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✏️ Solved Examples
Example 1 (Easy): √250 sabse paas ka integer.
15² = 225, 16² = 256 → 16 (6 dur vs 25 dur).
Check: 16² = 256 ≈ 250 ✅
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Example 2 (Medium): √5329 (perfect square hai).
70² = 4900, 80² = 6400 → 70–80. Aakhri 9 → 73 ya 77. 75² = 5625 > 5329 → 73.
Check: 73² = 4900 + 420 + 9 = 5329 ✅
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Example 3 (NCERT style): NCERT style: 500 sq m zameen, sabse bada integer side ka square bagicha?
22² = 484, 23² = 529 → 22 m.
Check: 484 ≤ 500 < 529 ✅
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Bijou ka jawab: “√250 lagbhag 16, par 16 se thoda kam!” Aur Akhil ka rumaal: 11 cm. 🎯
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🏠 Andaaza Kahan Kaam Aata
- 🧵 Darzi: kapde se sabse bada square tukda.
- 🏡 Plot: 300 sq yd ka square plot ≈ 17.3 yd side.
- 📺 Screen: area se side ka andaaza.
- 🧮 Exam check: √1000 ≈ 31.6; jawab 100 aaya to galat.
- 🌳 Bagicha: 200 paudhe square mein: 14 × 14 = 196, 4 bachenge.
Ghero, pass karo, aur dekho sawaal kaunsa integer maang raha.
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Perfect square (4 digit) | ✅ Ghera + aakhri digit + 5 wala square | Seedha jawab |
| Perfect square nahi, “lagbhag” | ✅ Nazdeeki integer | Kam doori wala |
| “Sabse bada fit” | ✅ Neeche wala integer | Upar wala fit nahi hoga |
| Decimal tak chahiye | ✅ 0.1 ke kadam | Ghera chhota karte jao |
| Aakhri digit 2, 3, 7, 8 | ❌ Integer root nahi | Sirf andaaza |
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❌→✅ Worked Mistake
Sawaal: 150 sq cm kapde se sabse bada integer side ka square?
❌ Galat: “√150 ≈ 12.2, paas ka 12… nahi 12² = 144 chalega. Par 13 bhi paas hai, 13 lo.”
✅ Sahi: 13² = 169 > 150, kapda kam padega → 12 cm. Check: 144 ≤ 150 ✅
🧠 Rule of thumb: Fit karna ho to hamesha neeche wala square; lagbhag ho to paas wala.
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📝 10 MCQs
Q1. √90 kin integers ke beech?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 81 < 90 < 100 ✅
- (b) galat: 64 aur 81 dono 90 se chhote.
- (c) galat: 100 aur 121 dono 90 se bade.
- (d) galat: Aadha karna root nahi hota.
Q2. √1936 = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 44² = 1936 ✅
- (b) galat: 46² = 2116, zyada hai.
- (c) galat: 34² = 1156, kam hai, isliye yeh galat.
- (d) galat: 54² = 2916, bahut zyada.
Q3. Square 9 pe khatam, root ka aakhri digit?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 3² = 9, 7² = 49 ✅
- (b) galat: 9² = 81, 1 pe khatam, isliye yeh galat.
- (c) galat: 1 ya 9 to 1 wale square ke.
- (d) galat: 7 bhi option hai, sirf 3 nahi.
Q4. √250 ka paas wala integer?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 256 sirf 6 dur ✅
- (b) galat: 225 to 25 dur hai, isliye yeh galat.
- (c) galat: 25² = 625, bahut zyada.
- (d) galat: Aadha karna root nahi hota.
Q5. 125 sq cm se sabse bada integer square rumaal?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 121 ≤ 125 ✅
- (b) galat: 144 > 125, fit nahi hoga.
- (c) galat: 225 to bahut bada hai.
- (d) galat: 10 fit hai, par 11 bada fit.
Q6. 45² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 4 × 5 | 25 = 2025 ✅
- (b) galat: 5 wale square 25 pe khatam.
- (c) galat: 16 to 4 × 4 se aata, galat.
- (d) galat: 21 ka koi aadhar nahi.
Q7. √3000 lagbhag?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 55² = 3025, sabse paas ✅
- (b) galat: 30² = 900, bahut kam, isliye yeh galat.
- (c) galat: 300² = 90000, bahut zyada.
- (d) galat: 17² = 289, bahut kam, isliye yeh galat.
Q8. √2 kin decimals ke beech?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 1.96 < 2 < 2.25 ✅
- (b) galat: 1.5² = 2.25 pehle se zyada.
- (c) galat: 1.2² = 1.44, abhi door.
- (d) galat: 2² = 4, bahut zyada, isliye yeh galat.
Q9. √7056 (perfect square)?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 80–90, 6 → 84/86, 85² = 7225 → 84 ✅
- (b) galat: 86 > 85, aur 85² pehle se zyada.
- (c) galat: 76² = 5776, kam hai, isliye yeh galat.
- (d) galat: 94² = 8836, zyada hai.
Q10. 200 paudhe square mein, kitne bachenge?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 14² = 196, 4 bache ✅
- (b) galat: 200 perfect square nahi.
- (c) galat: 225 ke liye 25 kam padte.
- (d) galat: 14 to qatar ki ginti hai.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Andaaza aur asli root mein kitna farak?
Integer andaaza 1 ke andar; decimal kadam se aur pass.
2. Aakhri digit 5 ho to?
Root 5 pe hi khatam.
3. Paas wala kaise tay karein?
Dono squares se doori ginti: kam doori wala.
4. √ fraction ka andaaza?
√(1/2) ≈ 0.7, kyunki 0.7² = 0.49.
5. Calculator kyun nahi?
Andaaza calculator ka jawab check karne ke liye bhi zaroori.
6. Aage kya?
Ab cube: Post 6 — Cube Numbers Aur Taxicab 1729.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy √50 kin integers ke beech?
Solution dekho
7 aur 8. Check: 49, 64 ✅
Q2. Easy √30 ka paas wala integer?
Solution dekho
5 (25 se 5, 36 se 6). Check ✅
Q3. Easy Square 4 pe khatam: root ke options?
Solution dekho
2 ya 8. Check ✅
Q4. Easy 65² = ?
Solution dekho
4225. Check: 6 × 7 | 25 ✅
Q5. Medium √2116 = ?
Solution dekho
40–50, 6 → 44/46, 45² = 2025 < 2116 → 46. Check ✅
Q6. Medium √3249 = ?
Solution dekho
50–60, 9 → 53/57, 55² = 3025 → 57. Check: 57² = 3249 ✅
Q7. Medium √600 lagbhag?
Solution dekho
24² = 576, 25² = 625 → 24 (24 dur vs 25 dur). Check ✅
Q8. Medium 300 sq m mein sabse bada integer square?
Solution dekho
17 m (289). Check: 18² = 324 ✅
Q9. NCERT style Kya 2032 square? Andaaza bhi.
Solution dekho
45² = 2025, 46² = 2116 → nahi, √2032 ≈ 45. Check ✅
Q10. NCERT style √8281 = ?
Solution dekho
90–100, 1 → 91/99, 95² = 9025 → 91. Check: 91² = 8281 ✅
Q11. NCERT style √10 ek decimal tak.
Solution dekho
3.1² = 9.61, 3.2² = 10.24 → 3.2 paas (0.24 vs 0.39). Check ✅
Q12. NCERT style 1000 bachche square mein: kam se kam kitne aur?
Solution dekho
32² = 1024 → 24. Check ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Kaunsa bada: √50 + √50 ya √100?
Solution dekho
2√50 ≈ 14.1 > 10. Check ✅
Q2. √99 aur √101: kitna farak lagbhag?
Solution dekho
9.95 aur 10.05 → ≈ 0.1. Check ✅
Q3. 5-digit perfect square jo 1 pe khatam aur 100 ke paas root?
Solution dekho
10201 = 101². Check ✅
Q4. Square farsh 150 sq ft, side ek decimal tak?
Solution dekho
12.2² = 148.84, 12.3² = 151.29 → ≈ 12.2. Check ✅
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🏆 Olympiad Corner
Q1. √(10⁶ + 1) lagbhag?
Solution dekho
1000 (thoda zyada).
Q2. Kitne integers ka √ 5 aur 6 ke beech?
Solution dekho
26 se 35: 10.
Q3. √12321?
Solution dekho
111.
Q4. √(0.5) ek decimal?
Solution dekho
0.7.
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💗 CharuMam Motivation
Bijou ne “nahi pata” ki jagah “lagbhag 16” kaha. Andaaza lagana bhi maths hai, aur zindagi mein sabse zyada kaam aata! 🌟
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🟢 Master Recap
- 💡 Chaabi: Root ka andaaza = number ko do nazdeeki perfect squares ke beech gherna: a² < y < b² to a < √y < b. Phir aakhri digit se options kam karo aur beech ka square (jaise 45²) nikaal ke interval aadha karo.
- a² < y < b² → a < √y < b.
- Aakhri digit se 2 options.
- Beech ka 5 wala square se interval aadha.
- Lagbhag → paas wala integer.
- Fit → neeche wala.
- Decimal tak: 0.1 ke kadam.
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👉 Aage Kya?
Square se cube ki taraf: Post 6 — Cube Numbers Aur Taxicab 1729.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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